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We show that any topologically transitive codimension-one Anosov flow on aclosed manifold is topologically equivalent to a smooth Anosov flow thatpreserves a smooth volume. By a classical theorem due to Verjovsky, any higherdimensional codimension-one Anosov flow is topologically transitive. Recently,Simic showed that any higher dimensional codimension-one Anosov flow thatpreserves a smooth volume is topologically equivalent to the suspension of anAnosov diffeomorphism. Therefore, our result gives a complete classification ofcodimension-one Anosov flow up to topological equivalence in higher dimensions.  In this second version, the order of the presentation of the proof is changedand some minor errors in the previous version is corrected.Comment: 27 pages, no figure

On invariant volumes of codimension-one Anosov flows and the Verjovsky conjecture